Friday, April 10, 2015

One Hot Topic: Law of Sines & Cosines

When I work with high school math teachers in my consulting life, I am always surprised by how often the topic of Law of Sines and Law of Cosines gets brought up.  Now, this may seem random, but it is often in the context of what content standards we must teach students, yet we don't have time for.  However, with the adoption of Common Core State Standards, you may notice that these are actually now "Plus Standards", which means that they will not actually be assessed.  So, my take away from this feeling of panic from teachers, is that they are finding the transition  to Common Core difficult, they are unsure of what they still need to teach from the previous standards, and what is it exactly that they can now take off of their plates.

This always sparks great discussion because the bigger conversation, in my opinion, is: "Is it really that important that we teach students a formula to memorize?"  Law of Sines and Cosines are not the only ideas or concepts that fall into this category, and I was guilty for many years of spending my classroom time lecturing students on topics that involved some nasty formula to memorize.  However, I have made a huge effort to change this culture in my classroom.

If you haven't seen Diana Laufenberg's TED Talk, you need to!  (Find it here!)  I love the way she talks about the role of school changing as technology becomes more available to our students and they no longer need to rely on their teacher for all the information; they can find it elsewhere.  I am not suggesting that there is no longer a place for the teacher in a child's education, it just looks different than it used to.

So, I tried something a little different this year.  My students had just returned from a week long spring break and before the vacation we had finished up Right Triangle Trigonometry.  So, as a review, I drew this picture on the board and asked them to find all of the missing parts:

The context was: you have a 12 ft ladder that is propped up against the side of your house and the bottom of the ladder is 5 ft away from your house.

Students proficiently used their trig tables that we had generated together (I blogged about that HERE) to accurately solve for the missing pieces.  All spirits were high and I was pleased to see that students comfortably fell back into the swing of school after a week away.

Then I threw them a curve ball...

Same situation, you've still got a 12 ft ladder that is 5 ft away from the base of the building, however, instead of this building being your house, it's the Leaning Tower of Pisa.  Students realized immediately that this was no longer a right triangle and they were curious to know how they might solve it.  "Can we use trig?  Can we use the Pythagorean Theorem?"  I told them that the angle in the bottom left was 60 degrees and that it was indeed NOT a right triangle, so our "traditional" methods for solving were unavailable.

I gave them a quick pep talk that essentially summarized Diana's TED Talk and let them know that my purpose for giving them this problem was to see what they could discover and research on their own.  I did not want them to come back with the values of the missing side and angles, instead, I wanted them to explore possible methods for how they might find the answers.  Also, I told them that I made up this problem, so don't bother Googling it directly.

My hopes were for them to think about:

  • What makes this problem different/the same as the first one?  
  • What question should I Google/ask?
  • How do I decide if what I've found is valid?
  • Is this method that I've found really applicable/appropriate?

I'll admit, I had no idea how the process would go and was mentally preparing myself for this to be a total flop.  But when they came back to class, my mind was blown...

When students returned to class, I reminded them that I didn't expect them to have found the answers for the missing values.  I wanted them to come to class with an idea of how to solve for the missing values and have a discussion about these ideas with their peers.

Within seconds of discussion I was hearing students say, "I found this thing called the Law of Sines that I think we can use here."  Another student said, "I found that, too!"  One student said, "I found the Law of Sines, but I also found the Law of Cosines and I think that is the one we should use based on the information given."

Are you frickin' kidding me?!?!

Now, not all students had this information.  Some hadn't looked at it at all, others used the Pythagorean theorem and right triangle trig properties, but there were a good handful of students who were on the right path and had hit the jackpot!  I took a quick lap around the room and was able to capture some images of what students had written down on their papers.  Check it out:

Trying to use some of our more "traditional" methods for solving non-right triangles

I almost fell over when I saw this written down.  Imagine how I felt when she started explaining it to her peers.
Note: when I came by to take the picture, she said, "Don't mind the raspberry juice that I spilled on my paper."
Perhaps she's onto something :)

This student admitted that she had found the formula on Wolfram Alpha and got stuck

I feel like the question mark says something about her understanding of the concept,
but she was able to plug in values correctly

Admittedly, students really were unsure about what the Law of Sines and Law of Cosines was all about, but with a little nudging, I was able to get them plugging things in, refining their research, and persevering in their problem solving.  It was fun to bounce back and forth between groups of students to help push them further in the problem or answer some questions that had arisen since my last visit. It was also exciting to hear them argue with each other and ask such thoughtful questions of their peers.

But the most beautiful part was that students were sincerely interested in finding the values.  The opportunity for them to take ownership of their learning and for them to discover what was important played a major role in their willingness to try.

My big ah-ha moment here was that students really are capable of finding things on their own, but it is still important for the teacher to be involved.  My students still needed me to answer specific questions and to help explain things along the way, but THEY were a huge part of that conversation and process, which was magical and something that I hope they will remember.  I know I certainly will.

I told them that my goal was for them to take away some bigger skills from this activity than just remembering a formula.  I admitted to them that when I was a math major in college, it was essential that I knew how to work with others and ask appropriate questions when solving a problem.  Without these skills, I would have sunk like a stone in many of my upper division math classes!

After about an hour of working on this problem, I told them that they could move on to the next activity that I had planned for the day.  The only groups that moved on were those who had found the missing side and angles of the triangle.  Other groups persevered and continued working.  I brought to their attention that they had just spent an hour working on a challenging math problem and for them to reflect on this feeling of wanting to finish.  I asked them to contrast this experience with approaching a challenging problem in their math classes of the past.  Many students nodded and smiled; they knew that this was something special.

One student said to me, "Yeah, I've never spent this much time on one single math problem and actually enjoyed it.  What have you done to my brain, Ms. Balli?"

I see it as this: I've helped them learn how to use their own brain and not rely on mine.  Isn't that what we want for all of our students?


  1. You nailed it with that last line! That is exactly what I hope for my students.
    Inspiring post Jessica! Thank for committing the time to share your experience.

  2. Thanks, Amy! Yes, teaching them "stuff" is great, but not as impactful as teaching them how to think critically and problem solve.

  3. I liked how you started with a familiar right triangle problem that included using trig, then extended it without first teaching the math concepts needed. Particularly nice that you had them shift their focus from finding answers to the problem to having them discover possible methods for how they might find the answers. It is magical when we can find the right situation where students take ownership of their learning and persevere because they are interested! A fascinating post – well done!!

    1. Thanks, Steve! It's so essential that students make connections in what they are learning and too often it's the teacher who is making the connections for them. I am super impressed with and proud of my students for persevering and taking on the challenge of learning new mathematics! It's kind of what I live for! :)